In this article, we study finite dynamical systems defined over
graphs, where the functions are applied asynchronously. Our goal is
to quantify and understand stability of the dynamics with respect to
the update sequence, and to relate this to structural properties of
the graph. We introduce and analyze three different notions of
update sequence stability, each capturing different aspects of the
dynamics. When compared to each other, these stability concepts
yield different conclusions regarding the relationship between
stability and graph structure, painting a more complete picture of
update sequence stability.